Question

The key lists the ordered pairs that represent the locations of trees in Mr. Lee's backyard. He wants to plant a maple tree at the location (6, 5).



Based on this information, which statement is true?


The maple tree will be located 1 unit east and 1 unit north of the pecan tree.
The maple tree will be located 1 unit east and 1 unit north of the pecan tree.

The maple tree will be located 3 units east and 6 units north of the oak tree.
The maple tree will be located 3 units east and 6 units north of the oak tree.

The maple tree will be located 4 units east and 1 unit south of the elm tree.
The maple tree will be located 4 units east and 1 unit south of the elm tree.

The maple tree will be located 5 units east and 5 units north of the oak tree

Answer 1

Step-by-step explanation:

First notice that there is no difference between the maple trees and the oak trees; we have only two types, birch trees and "non-birch" trees. (If you don't believe this reasoning, think about it. You could also differentiate the tall oak trees from the short oak trees, and the maple trees with many branches as opposed to those with few branches. Indeed, you could keep dividing until you have them each in their own category, but in the end it will not change the probability of the birch trees being near each other. That is, in the end, you multiply the numerator by the number of ways to arrange the oak and maple trees and you also multiply the denominator by the number of ways to arrange the oak and maple trees, making them cancel out.)

The five birch trees must be placed amongst the seven previous trees. We can think of these trees as 5 dividers of 8 slots that the birch trees can go in, making ${8\choose5} = 56$ different ways to arrange this.

There are ${12 \choose 5} = 792$ total ways to arrange the twelve trees, so the probability is $\frac{56}{792} = \frac{7}{99}$.

The answer is $7 + 99 = \boxed{106}$.